The question asks to determine the interval where the function $f(x) = x^3 - 6x^2 - 15x$ is decreasing ("turun pada interval"). To solve this, we need to follow these steps:

Step 1: Find the derivative of $f(x)$

The first derivative $f'(x)$ determines the slope of the function: $f'(x) = 3x^2 - 12x - 15$

Step 2: Solve $f'(x) = 0$ to find critical points

Set $f'(x) = 0$: $3x^2 - 12x - 15 = 0$

Divide through by 3: $x^2 - 4x - 5 = 0$

Factorize: $(x - 5)(x + 1) = 0$

The critical points are: $x = -1 \quad \text{and} \quad x = 5$

Step 3: Test the intervals determined by the critical points

The critical points divide the number line into three intervals: $(-\infty, -1)$, $(-1, 5)$, and $(5, \infty)$. Test the sign of $f'(x)$ in each interval to determine whether $f(x)$ is increasing or decreasing.

1. Interval $(-\infty, -1)$: Pick a test point, such as $x = -2$: $f'(-2) = 3(-2)^2 - 12(-2) - 15 = 3(4) + 24 - 15 = 21 > 0$ $f'(x) > 0$, so $f(x)$ is increasing in this interval.

2. Interval $(-1, 5)$: Pick a test point, such as $x = 0$: $f'(0) = 3(0)^2 - 12(0) - 15 = -15 < 0$ $f'(x) < 0$, so $f(x)$ is decreasing in this interval.

3. Interval $(5, \infty)$: Pick a test point, such as $x = 6$: $f'(6) = 3(6)^2 - 12(6) - 15 = 3(36) - 72 - 15 = 108 - 87 = 21 > 0$ $f'(x) > 0$, so $f(x)$ is increasing in this interval.

Step 4: Identify the interval where $f(x)$ is decreasing

From the analysis, $f(x)$ is decreasing in the interval $(-1, 5)$.

Final Answer:

The correct choice is: $\boxed{A. -1 < x < 5}$

Would you like a detailed explanation of any step? Here are some related questions you might find helpful:

1. How do you determine where a function is increasing or decreasing?
2. What is the importance of critical points in analyzing a function?
3. How do you factor quadratic equations like $x^2 - 4x - 5 = 0$?
4. Can the second derivative help confirm increasing or decreasing intervals?
5. What is the graphical interpretation of $f'(x)$ changing signs?

Tip: Always verify your derivative calculations and test values in each interval to avoid sign errors.